EVERYDAY MATH
Standard deviation and variance calculator
Calculate mean, variance and standard deviation for an arbitrary numeric dataset with sample and population results kept explicitly separate.
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No regional currency, unit or language profile is applied. This choice does not convert values or change formulas, results, routes or legal guidance.
Enter up to 200 finite values. Negative numbers, zero and decimals are supported.
Population standard deviation
28 values calculated locally.- Count
- 8
- Mean
- 5
- Sum of squared deviations
- 32
- Population variance
- 4
- Population standard deviation
- 2
- Sample variance
- 4.57142857
- Sample standard deviation
- 2.13808994
Deviation table
| Value | Deviation from mean | Squared deviation |
|---|---|---|
| 2 | -3 | 9 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 4 | -1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
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Verification and calculation details
- Purpose
- Standard deviation and variance calculator
- Variables and units
- Use the labelled fields and units shown in the calculator. Results use those units unless a conversion is stated.
- Formula and method
- The exact formula and worked example are shown below. Deterministic formula executed locally in the browser.
- Assumptions
- Only entered values and stated assumptions are included. Local fees, rules and professional requirements are excluded unless explicitly shown.
- Engine version
- 1.0.0
- Production verification baseline
- Release 63 · source
998803f - Verification run
release-v63-2026-09-20- Last verified
- — Automated release checks passing
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How this calculator works
For N observations, calculate the arithmetic mean and the sum of squared deviations Σ(x−mean)² using a numerically stable running update. Population variance divides that sum by N. Sample variance divides it by N−1 and therefore requires at least two observations. Standard deviation is the non-negative square root of the selected variance.
Variables and inputs
- Dataset
- Required numeric input. one to 200 finite values; comma, space, semicolon or line separated; each value between −1 trillion and 1 trillion.
- Primary interpretation
- Required numeric input. allowlisted selection: population or sample; sample statistics require at least two values.
How to interpret the result
The calculator returns count and mean, sum of squared deviations, population variance and standard deviation, sample variance and standard deviation, per-value deviation table. Read each output against the assumptions you entered, compare alternatives one variable at a time, and use the result as an estimate rather than a promise.
Common mistakes to avoid
- Mixing units or entering a rate as a decimal when the field requests a percentage.
- Rounding intermediate values instead of the final displayed result.
- Leaving out a cost, timing rule or local requirement listed in the assumptions below.
Practical uses
Use the standard deviation and variance calculator to check a real scenario, compare a changed assumption and explain the arithmetic before relying on a quote, specification or decision.
Worked example
For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5 and the squared deviations sum to 32. Population variance is 32/8=4 and population standard deviation is 2. Treating the same observations as a sample gives variance 32/7≈4.5714 and standard deviation ≈2.1381.
Assumptions and limitations
- every entered value belongs to the same variable and uses the same unit
- population mode means the entered observations are the complete population of interest
- sample mode estimates a larger population from the entered observations
- observations are included with equal weight
- the calculator does not decide whether sample or population statistics are appropriate
- standard deviation describes spread but does not establish distribution shape, causation or data quality
- variance uses the square of the observation unit
- displayed decimals are rounded after full-precision calculation
Important notes
Results are estimates based on the values entered. Confirm rates, prices, units and professional requirements that apply in your location. Review the Answer Atlas Calculation Standard and corrections policy for formula, privacy and correction rules.
Frequently asked questions
Should I use sample or population standard deviation?
Use population statistics when the entered values are every member of the population you want to describe. Use sample statistics when those values are a sample used to estimate a larger population.
Why does sample variance divide by N−1?
The N−1 denominator is the usual correction when estimating population variance from a sample whose mean was calculated from the same data.
Can I enter negative numbers and decimals?
Yes. Enter up to 200 finite values separated by commas, spaces, semicolons or line breaks. All values should describe the same variable in the same unit.